3 Coronal Magnetic Field Models

The magnetic fields observed in the solar photosphere extend out into the corona, where they structure the plasma, store free magnetic energy and produce a wide variety of phenomena. While the distribution and strength of magnetic fields are routinely measured in the photosphere, the same is not true for the corona, where the low densities mean that such measurements are very rare (Cargill, 2009). To understand the nature of coronal magnetic fields, theoretical models that use the photospheric observations as a lower boundary condition are required.

In this section, we survey the variety of techniques that have been developed to model the coronal magnetic field based on the input of photospheric magnetic fields. We limit our discussion to global models in spherical geometry, and do not consider, for example, models for the magnetic structure of single active regions. In future revisions of this review, we will extend our discussion to include observations of both coronal and prominence magnetic fields in order to determine the validity of these models. Applications of the global models are the subject of Section 4.

A difficulty arising with any such attempt at a global model based on observations of the photospheric magnetic field, is that we can only observe one side of the Sun at a single time (except for some limited observations by STEREO), but need data for all longitudes. This problem may be addressed either by compiling time series of full-disk observations into a synoptic observed magnetogram, or by assimilating individual active regions into a time-dependent surface flux transport model (Section 2.2). In this section, we will assume that the photospheric distribution of Br (R⊙, 𝜃,ϕ) at a given time has already been derived by one of these techniques, and consider how to construct the magnetic field of the corona.

Virtually all global models to date produce equilibria. This is reasonable since the coronal magnetic evolution on a global scale is essentially quasi-static. The magnetic field evolves through a continuous sequence of equilibria because the Alfvén speed in the corona (the speed at which magnetic disturbances propagate) is of the order of 1000 km s–1, much faster than the large-scale surface motions that drive the evolution (1 – 3 km s–1). To model this quasi-static evolution, nearly all models produce a series of independent “single-time” extrapolations from the photospheric magnetic fields at discrete time intervals, without regard to the previous magnetic connectivity that existed in the corona. The exception is the flux transport magneto-frictional model (Section 3.2) which produces a continuous sequence of equilibria. In this model, the equilibrium assumption is violated locally for brief periods during dynamical events such as CMEs or flares. Individual CME events are becoming practical to simulate in fully time-dependent MHD codes, but equilibrium models are still the only feasible way to understand when and where they are likely to occur.

We divide the techniques into four categories. The first three are potential field source surface models (Section 3.1), force-free field models (Section 3.2), and magnetohydrostatic models (Section 3.3). All of these models output only (or primarily) the magnetic field. The fourth category are full MHD models (Section 3.4), which aim to self-consistently describe both the magnetic field and other plasma properties.

 3.1 Potential field source surface models
 3.2 Nonlinear force-free field models
  3.2.1 Optimisation method
  3.2.2 Force-free electrodynamics method
  3.2.3 Flux transport and magneto-frictional method
 3.3 Magnetohydrostatic models
 3.4 Full magnetohydrodynamic models

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